LECTURE 4. CONTROL AND STABILIZATION OF BALANCE SYSTEMS 387- Since particles in constant magnetic fields move in circles with constant
speed, they have a sinusoidal time dependence, and hence so do the controls.
This has led to the "steering by sinusoids" approach in many nonholonomic
steering problems (see for example Murray and Sastry [1993]).
The equations for x and y are linear first order equations in the velocities and are
readily solved:
[
x(t)] [ cos(2.-\t) sin(2.-\t)] [±(0)]
y(t) - -sin(2.-\t) cos(2.-\t) y(O) ·
Integrating once more and using the initial conditions x(O) = 0, y(O) = 0 gives:[
x(t)] 1 [cos(2.-\t) - 1 sin(2.-\t) ] [-y(O)]
y(t) = 2.-\ -sin(2.-\t) cos(2.-\t) - 1 x(O) ·
The other boundary condition x(T) = 0, y(T) = 0 gives
,\ _ mr- T.
Using this information, we find z by integration: from i = xy-yx and the preceding
expressions, we get
1z(t) =
2
,\ [-±(0)^2 - y(0)^2 + cos(2At)(x(0)^2 + y(0)^2 )].
Integration from 0 to T and using z(O) = 0 gives
T
z(T) =
2,\ [-±(0)^2 - y(0)^2 ].Thus, to achieve the boundary condition z(T) = a, one chooses
x(0)2 + y(0)2 = - 2;~a.
One also finds that
1' T
~la [x(t)2
+ if(t)
2
] dt =~la [± (0)2
+ iJ(0)
2
] dt= ~ [±(0)2 + y(0)2]
7rna
T'so that the minimum is achieved when n = -1.
Summary. The solution of the optimal control problem is given by choosing initialconditions such that ±(0)^2 + y(0)^2 = 27ra/T^2 and with the trajectory in the xy
plane given by the circle[
x(t)] = 2 [cos(27rt/T) - 1 -sin(27rt/T) ] [-y(O)]
y(t) 2.-\ sin(27rt/T) cos(27rt/T) - 1 x(O)
and with z given byz(t) = ~ -ta^2 sin (
2
;t).
Notice that any such solution can be rotated about the z axis to obtain another
one.